Throughout a time interval, while the speed of a particle increases as it moves along x-axis, its velocity and acceleration might be

Options

A. positive and positive respectively
B. positive and negative respectively
C. negativetive and negative respectively
D. negative and positive respectively

Explanation

To solve the problem, we need to analyze the relationship between speed, velocity, and acceleration of a particle moving along the x-axis.

### Step-by-Step Solution:

1. **Understanding Speed and Velocity**:
- Speed is defined as the magnitude of velocity. Therefore, if the speed of a particle is increasing, it implies that the magnitude of its velocity is also increasing.

2. **Relationship Between Velocity and Acceleration**:
- The equation of motion states that the final velocity (v) can be expressed as:
\[ v = u + at \]
where \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time.

3. **Analyzing the Signs**:
- If the speed is increasing, it means that the velocity is changing in a way that its magnitude is increasing. This can happen in two scenarios:
- Both velocity and acceleration are positive (moving in the positive direction and speeding up).
- Both velocity and acceleration are negative (moving in the negative direction and speeding up).

4. **Conclusion**:
- Therefore, we can conclude that if the speed of a particle is increasing, its velocity and acceleration must either both be positive or both be negative. This leads us to the possible combinations:
- Positive velocity and positive acceleration.
- Negative velocity and negative acceleration.

5. **Identifying Correct Options**:
- Based on the analysis, the correct options for the signs of velocity and acceleration when speed is increasing are:
- Option A: Positive velocity and positive acceleration.
- Option C: Negative velocity and negative acceleration.

### Final Answer:
The correct options are A and C.